The statement of well ordering principle appears in different mode - on subsets of natural numbers, or well-ordering of every (non-empty) set. For the question below, I am considering it w.r.t. non-empty subsets of natural numbers.
The question below appeared in my mind, and I was unable to justify myself by referring various articles/books. It started from proof of Division Algorithm.
Let $a>b>0$ be integers. Then there are (unique) integers $q$ and $r$ such that $a=bq+r$ where $0\le r<q$.
Proof: Consider $S=\{ a-bn \,\,| \,\, n\in \mathbb{Z}_{\ge 0}, \,\, a-bn\ge 0\}.$ Then $S$ is non-empty (since $a\in S$) subset of $\mathbb{Z}_{\ge 0}$; by well-ordering principle, it contains least element, say $r=a-bq$. ..... [the rest of the proof goes for proving that $r$ satisfy above conditions.]
Question. Since $S$ is finite set, do we really need well-ordering principle in proof of Division Algorithm? To make more precise, let us make two statements.
i) A non-empty finite subset of $\mathbb{Z}_{\ge 0}$ contains least element.
ii) A non-empty infinite subset of $\mathbb{Z}_{\ge 0}$ contains least element.
So, in statement of well-ordering principle, do we need to include both i) and ii)?
(The inclusion of i) and ii) in well-ordering principle is done simply by ignoring the words finite/infinite in these statements.)
I was feeling that only ii) is the main part of Well-ordering principle, whereas i) also holds if $\mathbb{Z}_{\ge 0}$ is replaced by $\mathbb{Z}$ or $\mathbb{Q}$ or by $\mathbb{R}$. But, I am unable to clarify my doubts.
Can anyone clarify my doubts?